Theorem his52

Description: Associative law for inner product. (Contributed by NM, 13-Feb-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	his52	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 ·ih ( ( ∗ ‘ 𝐴 ) ·ℎ 𝐶 ) ) = ( 𝐴 · ( 𝐵 ·ih 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cjcl	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 𝐴 ) ∈ ℂ )
2		his5	⊢ ( ( ( ∗ ‘ 𝐴 ) ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 ·ih ( ( ∗ ‘ 𝐴 ) ·ℎ 𝐶 ) ) = ( ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) · ( 𝐵 ·ih 𝐶 ) ) )
3	1 2	syl3an1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 ·ih ( ( ∗ ‘ 𝐴 ) ·ℎ 𝐶 ) ) = ( ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) · ( 𝐵 ·ih 𝐶 ) ) )
4		cjcj	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) = 𝐴 )
5	4	oveq1d	⊢ ( 𝐴 ∈ ℂ → ( ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) · ( 𝐵 ·ih 𝐶 ) ) = ( 𝐴 · ( 𝐵 ·ih 𝐶 ) ) )
6	5	3ad2ant1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) · ( 𝐵 ·ih 𝐶 ) ) = ( 𝐴 · ( 𝐵 ·ih 𝐶 ) ) )
7	3 6	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐵 ·ih ( ( ∗ ‘ 𝐴 ) ·ℎ 𝐶 ) ) = ( 𝐴 · ( 𝐵 ·ih 𝐶 ) ) )